In this talk, we will show the rigidity of the Delaunay triangulated plane under Luo’s discrete conformality (also called the vertex scaling). More precisely, let T=(V,E,F) be a topological triangulation of the plane. Let l,l’ be two PL-metrics of T such that the induced distance structures are isometric to the plane. Suppose l and l’ are discrete conformal in the sense that there exists a function u on V such that l’_{ij}=e^{u_i+u_j}l_{ij}. We further assume l satisfies the Delaunay condition, l’ satisfies the uniformly Delaunay condition and both l and l’ satisfy the uniformly nondegenerate condition. Then l and l’ differ by a constant factor. This a joint work with Tianqi Wu.